01 SSetsets - National Book Foundation CHAPTER 1.pdfvenn diagram. 5. (i) A = All consonants of...
Transcript of 01 SSetsets - National Book Foundation CHAPTER 1.pdfvenn diagram. 5. (i) A = All consonants of...
Geometry
xBo
01
Learning Outcomes
Use language, notation and venn diagram to describe;
Set and its elements,
Finite and infinite set,
Empty set,
Singleton set,
Universal set.
(i)
(ii)
(iii)
(iv)
(v)
We often deal with groups or collection of objects in real life. In this chapter we will do mathematics with
“collection of objects”. Set is the mathematical way to represent collection or group of objects. Set are
used in every field of mathematics. They are important building blocks for more complex mathematical
structures.
SetsSets
SETS
06
In this world around us, things exist in collections. There are many
words to describe collection of objects, like a flock of birds, a pile of
books, a group of people, a team of players, bouquet of flowers, etc.
In mathematics, the term “set” is used to describe “a collection of
“well-defined” and distinct objects”.
07
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
Days of week.
Collection of distinct objects
Well-defined collection
Consider the following collection of days of week.
This is a collection of well-defined and distinct objects because from the
description “days of week” the reader is able to list all the elements of the
collection. No two objects in this collection are the same. Hence it is a set.
Consider the collection of best players of cricket team.
The collection of best players of cricket team is not well defined and hence not a
set.
Collection of distinctobjects
Best players of cricket team
Not a well definedcollection
every
one
may not
agree on
which
players
are the
best.
In general, a set is not any
collection of objects. The
objects in a set must be
well-defined and distinct.
eK y factIn a set
The word “distinct” means that the
objects of a set must all be different.
Elements cannot be repeated.
Order of elements does not matter.
Key fact
08
There are at least two common methods of specifying a set. One is to describe
the elements of a set in words and other is to list elements of the set. Sets are
usually represented by capital letters A, B, C,...
N = Natural numbers less than 10
N = {1, 2, 3, 4, 5, 6, 7, 8, 9}
B = vowels in english alphabets
B = {a, e, i,o, u}
C = even numbers less than 100
C = {0, 2, 4,...,98}
Each object in a set is called an element or member of the set. Elements of a set
can be symbols, numbers or objects etc. We relate a set and its element by using
the symbol �. In the following set
A = {1, 3, 5,..., 19}
Since 3 is is an element of A we write 3 �A,
and 2 is not an element of A we write 2 � A
K cey fa t
The symbol
� denotes “is an element of “
� denotes “is not an element of”
e aK y f ct
Some times, it may not be possible to list
all the elements of a set, in such case we
begin to list a few elements and then use
..., to show that pattern continues up to the
last element.
Let B be collection of all tall
boys in the class. Is B a set?
Ch ce k
Point
A is set of odd numbers less than 20.
List all the elements of A in set notation.
is 2 � A?
Ch cke
Point
State weather each of the following statement is true
or false, if it is false explain why.
P� {p, e, n} {P} � {p, e, n}
1 � set of even number Train � {t, r, a, i, n}
Ch cke
Point
We can represent a set using rectangle and circle. This geometrical
representation of a set is called venn diagram.
The set of natural numbers less than 10 are represented through venn diagram
as follows.
(i) N = All natural numbers less than 10
N = {1, 2, 3, 4, 5, 6, 7, 8, 9}.
N
1 2 3 4 5 6
7 8 9
(ii) U = Whole numbers less than 100
U = {0, 1, 2,..., 99}
(iii) E = Even numbers less than 100
E = {0, 2, 4,..., 98}
E
W
If all the elements of a set are listed one after another, and the process eventually
stops, such set is called finite set.
The number of elements in a finite set A is denoted by n (A).
X = natural numbers less than 12
X = {1, 2, 3,..., 11} is finite set
n (X) = 11
On the other hand when it is not possible to list out all the elements of a set, such a
set is called infinite set.
W = whole numbers
W = {0, 1, 2, 3,...,}
E = Even numbers
E = {0, 2, 4,...,}
we write infinite set by listing a first few elements and then putting three dots “...”
showing that the pattern continues and there are infinite more elements of the set
to come.
Finite and infinite set
fKey act
Infinite set never ends.
State number of elements in each of the following
sets.
Odd numbers that are divisible by 2.
Dinosaur living in earth.
Triangles having four vertices.
Vowels in the word “PAKISTAN”.
Even numbers between 10 and 12.
Checkt
Poin
09
EUEven
numbersless
than 100
Even numbers less than 100
are infinite sets
10
Empty Set
Consider the set B consisting of all human beings who visited mars, then B does
not contain any element. i.e. B = { }
A set which contains no element is called empty or null set. Empty set is denoted
by the symbol “�” (pronounced as “phi”).
The following sets are empty sets.
E = even numbers between 2 and 4 = ����{ }
B = human beings who visited Venus = � = { }
Key fact
Set of natural numbers less than
1 is an empty set.
Consider the set C consisting of natural number between 5 and 7, then
C = {6}, the set C contains exactly one element.
A set consisting of only one element is called singleton set.
Singleton set
Key fact
Set of whole numbers less than
1 is a singleton set.Key fact
The set {�} is not an empty set. It contains
one element, the symbol �.
Universal setThe universal set is the largest possible set of all
elements under consideration for specific situation.
There are 40 students in grade six in a school.
Let A be the set of students of grade 6 who play
hockey.
In above set A, we are looking for the students of
class six who play hockey, not the whole class six.
The students of grade six is the universal set for
this particular situation.
A universal set is denoted by U.
The venn diagram representation of above set is as
follows.
AU
Students of grade 6
Students ofgrade 6 whoplay hockey
State whether each of
the following
statements is true or
false.
W = Whole numbers
= {0, 1, 2}
{�} = �
n (�) = 0
4 � even numbers
i � {a, e, i, o, u}
Check
Point
11
U = Whole numbers less than 30
U = {0, 1, 2,..., 29}
A = Even number less than 30.
A = {0, 2, 4,..., 28}
B = Multiples of 5 less than 30.
B = {5, 10, 15, 20, 25}
are represented through venn diagram as follows.
51525
24681214
161822242628
13
7
9
11
13
1719
21
23
25
27
29
01020
U
A BThe overlapping region of two circles
represent common elements to both
sets i.e. 0, 10, 20.
U = Whole numbers less than or equal to 30.
U = {0, 1, 2, 3..., 30}
X = Even numbers less than or equal to 20.
X = {0, 2, 4,..., 20}
Y = Odd numbers less than or equal to 25.
Y = {1, 3, 5,..., 25}
are represented by venn diagram as follows.
UX Y
0246
161820
8101214
1357
19212325
911131517
22
24
26
27
28
29
30
12
Exercise 1.1
Which of the following collections are sets.1.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
All colours in rainbow.
Consonants of English alphabets.
Students in your class.
Very intelligent employees of a company.
Prime numbers less than 100.
Very hard working students in your class.
All the letters in the word mathematics.
All outcomes of flipping a coin.
All results of rolling a die.
Months of solar year.
Islamic month of fasting.
Is the following statements true?2.
(i) {2, 4, 6, 8,..., 100} = {2, 4, 6, 8,...} explain your answer
(ii) {1, 3, 5, 7} = {7, 3, 5, 1} explain your answer
Describe the following sets.3.
(i) A = {0, 2, 4, 6, 8}
(ii) B = {1, 3, 5, 7, 9}
(iii) X = {violet, indigo, blue, green, yellow, orange, red}
(iv) Y = {mercury, venus, earth, mars, jupiter, saturn, uranus, neptune}
(v) C = {potato, carrot, raddish, beat, sweet potato}
List the elements of the following sets.4.
(i) N = natural numbers greater than or equal to 20.
(ii) P = prime numbers between 10 and 30.
(iii) M = multiples of 5 greater than 5.
(iv) A = primary colours
(v) B = persons having ages 200 years and above.
13
Write the universal set for the following sets and represent them through
venn diagram.
5.
(i) A = All consonants of English alphabets.
(ii) B = Even numbers less than 10.
(iii) C = Sum of dots less than 8 appearing when two dice are rolled.
List the elements of the following sets and represent them through venn
diagram
U = Natural numbers less than or equal to 20,
X = Prime numbers greater than 5 and less than 20,
Y = Odd numbers less than or equal to 20.
6.
Fill in the blanks with � or �.7.
(i) 5 ________ {2, 4, 6,...}
(ii)
(iii)
1 ________ {0, 4, 6, 8}
2 ________ odd numbers less than 19.
(iv) 39 _______ sum of dots appearing when two dice are rolled.
100 _______ {2, 4, 6,...}(v)
6 ________ set of outcomes die is rolled.(viii)
A number divisible by 2 _________ set of whole numbers.(ix)
The month of Ramzan __________ the set of solar months.(x)
Grass ________ {carrot, potato, onion}
(vi)
(vii)
Blue ________ set of primary colours.
Let U = colours of rainbow
A = primary colours
Represent the sets through venn diagram
8.
14
Review Exercise 1
1. Encircle the correct option in the following statements.
(i) If A = Students in your class then A is a/an __________ set.
(a) (b) (c) (d)empty finite infinite none of these
(ii) B is the set of months of Eid then it is a/an _________ set.
(a) (b) (c) (d)empty finite infinite none of these
(iii) C = First 10 multiples of 3 then n (C) = _________ .
(a) (b) (c) (d)3 6 9 10
(iv) D = Students of your class of age 5 years is a/an _____ set.
(a) (b) (c) (d)empty singleton infinite universal
(v) E = Multiples of 4 is a/an ________ set.
(a) (b) (c) (d)empty singleton finite infinite
(vi) Set of colours of rainbow has _______ elements.
(a) (b) (c) (d)6 5 7 9
(vii) Set of triangles is a/an __________ set.
(a) (b) (c) (d)empty singletonfiniteinfinite
(viii) Set of even number divisible by 3 is a/an ________ set.
(a) (b) (c) (d)empty finite infinite none of these
(ix)
(a)
Set of natural numbers between 1 and 3 is a/an _________ set.
(b) (c) (d)infinite singleton empty none of these
(x)
(a)
The universal set U for a set of even numbers is set of ________
numbers.
(b) (c) (d)natural prime whole numbersodd
2. Represents the following sets through venn diagram.
(i) U = {1, 2, 3,..., 50},
A = multiples of 3 less than 50,
B = multiples of 5 less than or equal to 50.
15
(ii) U = {1, 2, 3,..., 40},
A = {1, 2, 3,..., 20},
B = {21, 22, 23,..., 40}.
Represent the following sets through venn diagram
U = {0, 1, 2, ... , 10}
A = {0, 2, 4, 6, 8}
B = {1, 2, 3, 4, 6, 7, 9}
3.
Take any two sets of your choice, write their universal set. Make a venn
diagram using.
(i) Two circles for each set.
(ii) What do you guess the shared portion of circles represents?
4.
SetWell definedDistinctVenn diagramFinite set
Infinite setEmpty setSingleton setUniversal set
Words Board
Set is a collection of well defined and distinct objects.
Venn diagram is a geometrical representation of sets.
A set consisting of limited number of elements is called a finite set, otherwise it
is called infinite set.
A set having no element is called empty set.
A set consisting of only one element is called singleton set.
A universal set is the largest possible set of all elements under consideration
for a specific situation.
In venn diagram the overlapping region of two circles represents the
elements which are common to both sets.
I have learnt